1. Basic Definition of a Factor Model

Factor models of security returns decompose the random return on each of a cross-section of assets into factor-related and asset-specific returns. Letting r denote the vector of random returns on n assets, and assuming k factors, a factor decomposition has the form: r = a+Bf + ε (1) where B is a nxk-matrix of factor betas, f is a random k−vector of factor returns, and ε is an n−vector of asset-specific returns. The n−vector of coefficients a is set so that E[ε] = 0. By defining B as the least squares projection B = cov(r, f)C−1f, it follows that cov(f, ε) = 0 kxn. The factor decomposition (1) puts no empirical restrictions on returns beyond requiring that the means and variances of r and f exist. So in this sense it is empty of empirical content. To add empirical structure it is commonly assumed that the asset-specific returns ε are cross-sectionally uncorrelated, E[εε0] = D where D is a diagonal matrix. This implies that the covariance matrix of returns can be written as the sum of a matrix of rank k and a diagonal matrix: cov(r, r0) = Bcov(f, f 0)B0 +D. (2) This is called a strict factor model. Without loss of generality one can assume that cov(f, f 0) has rank k, since otherwise one of the factors can be removed (giving a k − 1 factor model) without affecting the fit of the model